Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds

Optimal Principal Component Analysis in Distributed and Streaming Models

We study the Principal Component Analysis (PCA) problem in the distributed and streaming models of computation. Given a matrix $A \in R^{m \times n},$ a rank parameter $k < rank(A)$, and an accuracy parameter $0 < ε< 1$, we want to output an $m \times k$ orthonormal matrix $U$ for which $$ || A - U U^T A ||_F^2 \le \left(1 + ε\right) \cdot || A - A_k||_F^2, $$ where $A_k \in R^{m \times n}$ is the best rank-$k$ approximation to $A$. This paper provides improved algorithms for distributed PCA and streaming PCA.

Randomized Algorithms for Tracking Distributed Count, Frequencies, and Ranks

We show that randomization can lead to significant improvements for a few fundamental problems in distributed tracking. Our basis is the {\em count-tracking} problem, where there are $k$ players, each holding a counter $n_i$ that gets incremented over time, and the goal is to track an $\eps$-approximation of their sum $n=\sum_i n_i$ continuously at all times, using minimum communication. While the deterministic communication complexity of the problem is $Θ(k/\eps \cdot \log N)$, where $N$ is the final value of $n$ when the tracking finishes, we show that with randomization, the communication c…

工程化可扩展的分布式列表排名

Engineering Scalable Distributed List Ranking

通过改进Sibeyn的稀疏统治集算法，利用间接通信、输入局部性和消息合并，实现在多达24,576个核心上对数十亿元素进行可扩展的列表排名。

A Generalized Randomized Rank-Revealing Factorization

We introduce a Generalized Randomized QR-decomposition that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a communication-optimal spectral divide-and-conquer algorithm for the nonsymmetric eigenvalue problem. In this paper, we establish that this randomized QR-factorization satisfies the strong rank-revealing properties. We also formally prove its stability, making it suitable in applications. Finally, we present numerical experiments which demonstrate that our theore…

用于分布式架构上低秩近似的块子采样随机Hadamard变换

Block subsampled randomized Hadamard transform for low-rank approximation on distributed architectures

本文提出由子采样随机Hadamard变换组合而成的块SRHT矩阵，证明其作为无影子空间嵌入的性质，并将其应用于分布式架构下大规模矩阵的低秩近似计算。